Here is a very simple tree with two branches:

Two-branch tree

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These are the steps that a simple computer program follows to draw the tree, with a red arrow indicating where the computer’s focus is at each stage:

Two-branch tree stage 1

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2-Tree stage 2

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2-Tree stage 3

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2-Tree stage 4

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2-Tree (animated)

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If you had to give the computer an explicit instruction at each stage, the instructions might look something like this:

1. Start at node 1, draw a left branch to node 2 and colour the node green.
2. Return to node 1.
3. Draw a right branch to node 3 and colour the node green.
4. Finish.

Now try a slightly less simple tree with branches that fork twice:

Four-branch tree (static)

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These are the steps that a simple computer program follows to draw the tree, with a red arrow indicating where the computer’s focus is at each stage:

4-Tree #1

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4-Tree #2

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4-Tree #3

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4-Tree #4

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4-Tree #5

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4-Tree #6

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4-Tree #7

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4-Tree #8

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4-Tree #9

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4-Tree #10

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4-Tree #11

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4-Tree (animated)

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If you had to give the computer an explicit instruction at each stage, the instructions might look something like this:

1. Start at node 1 and draw a left branch to node 2.
2. Draw a left branch to node 3 and colour it green.
3. Return to node 2.
4. Draw a right branch to node 4 and colour it green.
5. Return to node 2.
6. Return to node 1.
7. Draw a right branch to node 5.
8. Draw a left branch to node 6.
9. Draw a left branch to node 7 and colour it green.
10. Return to node 6.
11. Draw a left branch to node 8 and colour it green.
12. Finish.

It’s easy to see that the list of instructions would be much bigger for a tree with branches that fork three times, let alone four times or you. But you don’t need to give a full set of explicit instructions: you can use a program, or a list of instructions using variables. Suppose the tree has branches that fork f times. If f = 4, you will need an array variable level() with four values, level(1), level(2), level(3) and level(4). Now follow these instructions:

1. li = 1, level(1) = 0, level(2) = 0, ... level(f+1) = 0
2. level(li) = level(li) + 1
3. If level(li) = 1, draw a branch to the left and jump to step 7
4. If level(li) = 2, draw a branch to the right and jump to step 7
5. li = li - 1 (note that this line is reached if the tests fail in lines 3 and 4)
6. If li > 0, jump to step 2, otherwise jump to step 11
7. If li = f, draw a green node and jump to step 5
9. li = li + 1
10. Jump to step 2
11. Finish.

By changing the value of f, a computer can use those eleven basic instructions to draw any size of tree (I’ve left out details like changes in the length of branches and so on). When f = 4, the tree will look like this:

16-Tree (static)

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16-Tree (animated)

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With simple adjustments, the program can be used for other shapes whose underlying structure can be represented symbolically as a tree. The program is in fact a fractalizer, that is, it draws a fractal. So if you use a version of the program to draw fractals based on right-triangles, you can say you are “tright treeing” (tright = triangle-that-is-right).

Here is some tright treeing. Start with a simple isoceles right-triangle. It can be divided into smaller isoceles right-triangles by finding the midpoint of the hypotenuse, then repeating:

Right-triangle rep-2 stage 1

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Right-triangle #2

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Tright #3

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Tright #4

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Tright #5

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Tright #6

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Tright #7

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Tright #7 (no internal lines)

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You can distort the isoceles right-triangle in interesting ways by finding the midpoint of a side other than the hypotenuse, like this:

Right-triangle (distorted) #1

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Distorted tright #2

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Distorted tright #3

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Distorted tright #4

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Distorted tright #5

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Distorted tright #6

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Distorted tright #7

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Distorted tright #8

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Distorted tright #9

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Distorted tright #10

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Distorted tright #11

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Distorted tright #12

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Distorted tright #13

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Distorted tright (animated)

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Here’s a different right-triangle. When you divide it regularly, it looks like this:

Right-triangle rep-3 stage 1

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Rep-3 Tright #2

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3-Tright #3

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3-Tright #4

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3-Tright #5

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3-Tright #6

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3-Tright #7

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3-Tright #8

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3-Tright #9

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3-Tright (one colour)

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When you distort the divisions, you can create interesting fractals (click on images for larger versions):

Distorted 3-Tright

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Distorted 3-Tright

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Distorted 3-Tright

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Distorted 3-Tright

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Distorted 3-Tright

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Distorted 3-Tright

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Distorted 3-Tright (animated)

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And when four of the distorted right-triangles (rep-2 or rep-3) are joined in a diamond, you can create shapes like these:

Creating a diamond #1

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Creating a diamond #2

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Creating a diamond #3

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Creating a diamond #4

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Creating a diamond (animated)

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Rep-3 right-triangle diamond (divided)

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Rep-3 right-triangle diamond (single colour)

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Distorted rep-3 right-triangle diamond

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Distorted 3-tright diamond

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Distorted 3-tright diamond

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Distorted 3-tright diamond

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Distorted 3-tright diamond

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Distorted 3-tright diamond

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Distorted 3-tright diamond

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Distorted 3-tright diamond

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Distorted 3-tright diamond

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Distorted 3-tright diamond

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Distorted 3-tright diamond

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Distorted 3-tright diamond (animated)

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Distorted rep-2 right-triangle

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Distorted 2-tright diamond

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Distorted 2-tright diamond

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Distorted 2-tright diamond

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Distorted 2-tright diamond

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Distorted 2-tright diamond (animated)

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